Universal quadratic forms and class numbers (UFOCLAN)

We focus on universal quadratic forms over number fields. They are well-studied, but their theory is far from complete even in the basic case of real quadratic fields and the available results are still sparser in higher degrees. Our approach combines geometric and analytic techniques. On the one hand, this comprises the geometry of quadratic lattices, their minimal vectors and indecomposable algebraic integers. From the other side, we consider their connection with class numbers in families of number fields via analytic tools.

Team leader & members

• Assoc. Prof. Vítězslav Kala
• approx. 5 postdocs and 3 Ph.D. students (see project web for details)

Selected publications

• Kala, P. Yatsyna, Lifting problem for universal quadratic forms, Adv. Math. 377 (2021), 107497
• Kala, M. Tinková, Universal quadratic forms, small norms and traces in families of number fields, Int. Math. Res. Not. IMRN (2023)
• Cherubini, A. Fazzari, A. Granville, V. Kala, P. Yatsyna, Consecutive real quadratic fields with large class numbers, Int. Math. Res. Not. IMRN (2023)

Contact

Charles University
Faculty of Mathematics and Physics
Mgr. Vítězslav Kala, Ph.D.
E: kala@karlin.mff.cuni.cz
T: (+420) 951 553 238

More information

We plan to accept a new Ph.D. student with starting date in October 2023. Please see the project webpage for details:
https://www1.karlin.mff.cuni.cz/~kala/web/ufoclan

We have an interest in motivated post graduate students interested in number theory.

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